Preprint\(p\)JonesWenzl idempotentsGaston Burrull, Nicolas Libedinsky and Paolo SentinelliAbstractFor a prime number \(p\) and any natural number \(n\) we introduce, by giving an explicit recursive formula, the \(p\)JonesWenzl projector \({}^p\!\operatorname{JW}_n\), an element of the TemperleyLieb algebra \(TL_n(2)\) with coefficients in \({\mathbf F}_p\). We prove that these projectors give the indecomposable objects in the \(\tilde{A}_1\)Hecke category over \({\mathbf F}_p\), or equivalently, they give the projector in \(\mathrm{End}_{\mathrm{SL}_2(\overline{{\mathbf F}_p})}(({\mathbf F}_p^2)^{\otimes n})\) to the top tilting module. The way in which we find these projectors is by categorifying the fractal appearing in the expression of the \(p\)canonical basis in terms of the KazhdanLusztig basis for \(\tilde{A}_1\). AMS Subject Classification: Primary 20G05; secondary 05E10.
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